Violation of Heisenberg’s Uncertainty Principle ()

Koji Nagata^{1}, Tadao Nakamura^{2}

^{1}Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon, Korea.

^{2}Department of Information and Computer Science, Keio University, Yokohama, Japan.

**DOI: **10.4236/oalib.1101797
PDF
HTML XML
1,428
Downloads
3,990
Views
Citations

Recently, violation of Heisenberg’s uncertainty relation in spin
measurements is discussed [J. Erhart* et al.*, Nature Physics 8, 185
(2012)] and [G. Sulyok *et al.*, Phys. Rev. A 88, 022110 (2013)]. We
derive the optimal limitation of Heisenberg’s uncertainty principle in a
specific two-level system (e.g., electron spin, photon polarizations, and so
on). Some physical situation is that we would measure **σ**_{x} and **σ**_{y}, simultaneously. The
optimality is certified by the Bloch sphere. We show that a violation of
Heisenberg’s uncertainty principle means a violation of the Bloch sphere in the
specific case. Thus, the above experiments show a violation of the Bloch sphere
when we use ±1 as measurement outcome. This conclusion agrees with recent researches
[K. Nagata, Int. J. Theor. Phys. 48, 3532 (2009)] and [K. Nagata* et al.*,
Int. J. Theor. Phys. 49, 162 (2010)].

Share and Cite:

Nagata, K. and Nakamura, T. (2015) Violation of Heisenberg’s Uncertainty Principle. *Open Access Library Journal*, **2**, 1-6. doi: 10.4236/oalib.1101797.

**Subject Areas:**** Applied Physics**

1. Introduction

The quantum theory (cf. [1] - [5] ) gives accurate and at-times-remarkably accurate numerical predictions. Much experimental data has fit to quantum predictions for long time.

As for the foundations of the quantum theory, Leggett-type non-local variables theory [6] is experimentally investigated [7] - [9] . The experiments report that the quantum theory does not accept Leggett-type non-local variables interpretation.

As for the applications of the quantum theory, the implementation of a quantum algorithm to solve Deutsch’s problem [10] on a nuclear magnetic resonance quantum computer was reported firstly [11] . An implementation of the Deutsch-Jozsa algorithm on an ion-trap quantum computer is also reported [12] . There are several attempts to use single-photon two-qubit states for quantum computing. Oliveira et al. implemented Deutsch’s algorithm with polarization and transverse spatial modes of the electromagnetic field as qubits [13] . Single- photon Bell states are prepared and measured [14] . Also the decoherence-free implementation of Deutsch’s algorithm was reported by using such single-photon and by using two logical qubits [15] . More recently, a one- way based experimental implementation of Deutsch’s algorithm was reported [16] .

In quantum mechanics, the uncertainty principle is any of the variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle known as complementary variables, such as its position x and momentum p, can be known simultaneously. For instance, in 1927, Werner Heisenberg stated that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa [17] . The formal inequality relating the standard deviation of position and the standard deviation of momentum was derived by Earle Hesse Kennard [18] later that year and by Hermann Weyl [19] in 1928.

Recently, Ozawa discusses universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement [20] . And experimental demonstration of a universally valid error-disturbance uncertainty relation in spin-measurements is discussed [21] . Violation of Heisenberg’s error-disturbance uncer- tainty relation in neutron spin measurements is also discussed [22] .

In this paper, we derive the optimal limitation of Heisenberg’s uncertainty principle in a specific two-level system (e.g., electron spin, photon polarizations, and so on). Some physical situation is that we would measure and, simultaneously. The optimality is certified by the Bloch sphere. We show that a violation of Heisenberg’s uncertainty principle means a violation of the Bloch sphere in the specific case. Especially, the Schrödinger uncertainty relation is equivalent to the Bloch sphere relation in the physical situation. Thus, the experiments [21] [22] show a violation of the Bloch sphere when we use as measurement outcome. This conclusion agrees with recent researches [23] [24] .

2. The Schrödinger Uncertainty Relation

In this section, we review the Schrödinger uncertainty relation. The derivation shown here incorporates and builds off of those shown in Robertson [25] , Schrödinger [26] and standard textbooks such as Griffiths [27] .

For any Hermitian operator, based upon the definition of variance, we have

(1)

We let and thus

(2)

Similarly, for any other Hermitian operator in the same state

(3)

for.

The product of the two deviations can thus be expressed as

(4)

In order to relate the two vectors and, we use the Cauchy-Schwarz inequality [28] which is defined as

(5)

and thus Equation (4) can be written as

(6)

Since is in general a complex number, we use the fact that the modulus squared of any complex number z is defined as, where is the complex conjugate of z. The modulus squared can also be expressed as

(7)

We let and and substitute these into the equation above to get

(8)

The inner product is written out explicitly as

(9)

and using the fact that and are Hermitian operators, we find

(10)

Similarly it can be shown that.

For a pair of operators and, we may define their commutator as.

Thus we have

(11)

and

(12)

where we have introduced the anticommutator,.

We now substitute the above two equations above back into Equation (8) and get

(13)

Substituting the above into Equation (6) we get the Schrödinger uncertainty relation

(14)

3. Main Result

In this section, we present an example that the Schrödinger uncertainty relation is optimal. The optimality is certified by the Bloch sphere. In fact, a violation of the Schrödinger uncertainty relation means a violation of the Bloch sphere in the specific case. We derive the Schrödinger uncertainty relation by using the Bloch sphere

relation in the specific case. Let be the variance of, i.e.,.

Statement 1

(15)

Proof. By using, we have

(16)

Thus,

(17)

QED

We define N as follows:

(18)

We define S as follows:

(19)

We discuss the relation between N and S in the following statement.

Statement 2

(20)

Proof. We have the following relation:

(21)

QED

Thus the Schrödinger uncertainty relation is optimal in the specific case. The optimality is certified by the Bloch sphere. A violation of the Schrödinger uncertainty relation means a violation of the Bloch sphere in the specific case. Thus, the experiments [21] [22] show a violation of the Bloch sphere when we use as measurement outcome. This conclusion agrees with recent researches [23] [24] .

4. Conclusions

In conclusions, we have derived the optimal limitation of Heisenberg’s uncertainty principle in a specific two- level system (e.g., electron spin, photon polarizations, and so on). Some physical situation has been that we would measure and, simultaneously. The optimality has been certified by the Bloch sphere. We have shown that a violation of Heisenberg’s uncertainty principle means a violation of the Bloch sphere in the specific case. Thus, the experiments [21] [22] have shown a violation of the Bloch sphere when we use as measurement outcome. This conclusion has agreed with recent researches [23] [24] .

It is very interesting to study whether Heisenberg’s uncertainty principle would violate when we would use a new measurement theory [24] as measurement outcome.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | Sakurai, J.J. (1995) Modern Quantum Mechanics. Addison-Wesley Publishing Company. |

[2] | Peres, A. (1993) Quantum Theory: Concepts and Methods. Kluwer Academic, Dordrecht. |

[3] | Redhead, M. (1989) Incompleteness, Nonlocality, and Realism. 2nd Edition, Clarendon Press, Oxford. |

[4] | von Neumann, J. (1955) Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton. |

[5] | Nielsen, M.A. and Chuang, I.L. (2000) Quantum Computation and Quantum Information. Cambridge University Press, Cambridge. |

[6] |
Leggett, A.J. (2003) Nonlocal Hidden-Variable Theories and Quantum Mechanics: An Incompatibility Theorem. Foundations of Physics, 33, 1469-1493. http://dx.doi.org/10.1023/A:1026096313729 |

[7] |
Gröblacher, S., Paterek, T., Kaltenbaek, R., Brukner, Č., Żukowski, M., Aspelmeyer, M. and Zeilinger, A. (2007) An Experimental Test of Non-Local Realism. Nature (London), 446, 871-875. http://dx.doi.org/10.1038/nature05677 |

[8] |
Paterek, T., Fedrizzi, A., Gröblacher, S., Jennewein, T., Żukowski, M., Aspelmeyer, M. and Zeilinger, A. (2007) Experimental Test of Nonlocal Realistic Theories without the Rotational Symmetry Assumption. Physical Review Letters, 99, Article ID: 210406. http://dx.doi.org/10.1103/PhysRevLett.99.210406 |

[9] |
Branciard, C., Ling, A., Gisin, N., Kurtsiefer, C., Lamas-Linares, A. and Scarani, V. (2007) Experimental Falsification of Leggett’s Nonlocal Variable Model. Physical Review Letters, 99, Article ID: 210407. http://dx.doi.org/10.1103/PhysRevLett.99.210407 |

[10] |
Deutsch, D. (1985) Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer. Proceedings of the Royal Society of London. Series A, 400, 97. http://dx.doi.org/10.1098/rspa.1985.0070 |

[11] |
Jones, J.A. and Mosca, M. (1998) Implementation of a Quantum Algorithm on a Nuclear Magnetic Resonance Quantum Computer. The Journal of Chemical Physics, 109, 1648. http://dx.doi.org/10.1063/1.476739 |

[12] |
Gulde, S., Riebe, M., Lancaster, G.P.T., Becher, C., Eschner, J., Häffner, H., Schmidt-Kaler, F., Chuang, I.L. and Blatt, R. (2003) Implementation of the Deutsch-Jozsa Algorithm on an Ion-Trap Quantum Computer. Nature, 421, 48-50. http://dx.doi.org/10.1038/nature01336 |

[13] |
de Oliveira, A.N., Walborn, S.P. and Monken, C.H. (2005) Implementing the Deutsch Algorithm with Polarization and Transverse Spatial Modes. Journal of Optics B: Quantum and Semiclassical Optics, 7, 288-292. http://dx.doi.org/10.1088/1464-4266/7/9/009 |

[14] | Kim, Y.-H. (2003) Single-Photon Two-Qubit Entangled States: Preparation and Measurement. Physical Review A, 67, Article ID: 040301(R). |

[15] |
Mohseni, M., Lundeen, J.S., Resch, K.J. and Steinberg, A.M. (2003) Experimental Application of Decoherence-Free Subspaces in an Optical Quantum-Computing Algorithm. Physical Review Letters, 91, Article ID: 187903. http://dx.doi.org/10.1103/PhysRevLett.91.187903 |

[16] |
Tame, M.S., Prevedel, R., Paternostro, M., Böhi, P., Kim, M.S. and Zeilinger, A. (2007) Experimental Realization of Deutsch’s Algorithm in a One-Way Quantum Computer. Physical Review Letters, 98, Article ID: 140501. http://dx.doi.org/10.1103/PhysRevLett.98.140501 |

[17] |
Heisenberg, W. (1927) über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik, 43, 172-198. http://dx.doi.org/10.1007/BF01397280 |

[18] |
Kennard, E.H. (1927) Zur Quantenmechanik einfacher Bewegungstypen. Zeitschrift für Physik, 44, 326-352. http://dx.doi.org/10.1007/BF01391200 |

[19] | Weyl, H. (1928) Gruppentheorie und Quantenmechanik. Hirzel, Leipzig. |

[20] |
Ozawa, M. (2003) Universally Valid Reformulation of the Heisenberg Uncertainty Principle on Noise and Disturbance in Measurement. Physical Review A, 67, Article ID: 042105. http://dx.doi.org/10.1103/PhysRevA.67.042105 |

[21] |
Erhart, J., Sponar, S., Sulyok, G., Badurek, G., Ozawa, M. and Hasegawa, Y. (2012) Experimental Demonstration of a Universally Valid Error-Disturbance Uncertainty Relation in Spin Measurements. Nature Physics, 8, 185-189. http://dx.doi.org/10.1038/nphys2194 |

[22] |
Sulyok, G., Sponar, S., Erhart, J., Badurek, G., Ozawa, M. and Hasegawa, Y. (2013) Violation of Heisenberg’s Error-Disturbance Uncertainty Relation in Neutron-Spin Measurements. Physical Review A, 88, Article ID: 022110. http://dx.doi.org/10.1103/PhysRevA.88.022110 |

[23] |
Nagata, K. (2009) There Is No Axiomatic System for the Quantum Theory. International Journal of Theoretical Physics, 48, 3532-3536. http://dx.doi.org/10.1007/s10773-009-0158-z |

[24] |
Nagata, K. and Nakamura, T. (2010) Can von Neumann’s Theory Meet the Deutsch-Jozsa Algorithm? International Journal of Theoretical Physics, 49, 162-170. http://dx.doi.org/10.1007/s10773-009-0189-5 |

[25] |
Robertson, H.P. (1929) The Uncertainty Principle. Physical Review, 34, 163-164. http://dx.doi.org/10.1103/PhysRev.34.163 |

[26] | Schrödinger, E. (1930) Zum Heisenbergschen Unscharfeprinzip. Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, 14, 296-303. |

[27] | Griffiths, D. (2005) Quantum Mechanics. Pearson Prentice Hall, Upper Saddle River. |

[28] |
Riley, K.F., Hobson, M.P. and Bence, S.J. (2006) Mathematical Methods for Physics and Engineering. Cambridge University Press, Cambridge, 246. http://dx.doi.org/10.1017/CBO9780511810763 |

Journals Menu

Contact us

+1 323-425-8868 | |

customer@scirp.org | |

+86 18163351462(WhatsApp) | |

1655362766 | |

Paper Publishing WeChat |

Copyright © 2024 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.